Graph searching and applications

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Cops-and-robbers games are useful in modelling various graph widths, which in turn are useful in the design of efficient algorithms. They are also useful in robotics by modelling different problems that robots used in practical situation may encounter. This proposal is concerned with the mathematics of the latter applications. In this case, a cop and a robber play on a given graph, by alternating moves from a vertex to one of its neighbours, until the cop occupies the same vertex as the robber. Thus a robot can attempt to locate an intruder in a network or in a maze. In particular, we investigate, for graphs on which the cop has a winning strategy, the length of games in which each player moves intelligently, that is, the cop tries to catch the robber as fast as possible while the robber tries to survive as long as possible. We also consider a new and more general game, in which a number of cops play on one graph G, trying to capture a number of robbers on another graph H. The capture relation between G and H governs the game. This can model, for example, a robot moving in one room, while his position is translated to a situation elsewhere (a discrete version of a mouse on a pad one uses with a computer, except of course the robot is not controlled by a human). As an extension, we look at the games on infinite graphs since their gaming properties are quite different. While this is a purely mathematical endeavour, some of the results could be applicable to the web graph as some people like to model it by an in infinite one. Note that the numerous people that work on graph searching are ready for industrial collaboration.